Under this assumptions it was proved in [1]}, that \(A\in \Psi ^m_{\rho ,\delta }(G)\)  if and only if its matrix-valued symbol \(\sigma _A\)  satisfies the symbol estimates
\(\Vert \partial _x^\beta \mathbb {D}^\alpha \sigma _A(x,\xi )\Vert _{\textnormal {End}(\mathbb {C}^{d_\xi })}\le C_{\alpha ,\beta }\langle \xi \rangle ^{m-\rho |\alpha |+\delta |\beta |},\, (x,\xi )\in G\times \widehat{G}.\) 
